1. Field of the Invention
The present invention relates to a method for measuring relative displacements among a plurality of objects with a high degree of accuracy by utilizing diffraction and interference phenomena of waves through diffraction gratings formed on the objects.
2. Description of the Prior Art
As a prior art, a method for measuring a relative displacement by utilizing diffraction gratings disclosed by Flanders et al. in Applied Phisics Letters 31,426 (1977) will be described with reference to FIG. 1. Diffraction gratings G1 and G2 having the same period d are formed on two objects 1 and 2, respectively, and are spaced apart from each other in parallel with each other. When the wave I is vertically incident to the diffraction grating G1, it is diffracted by an angle .theta. (d sin .theta.=n.lambda., where n is an integer representing the order of diffraction) which is dependent upon the period d and the wavelength .lambda. of the wave I. An intensity of the diffracted waves varies in response to a relative position between the diffraction gratings G1 and G2 so that the relative displacement between the objects 1 and 2 or between the diffraction gratings G1 and G2 can be measured by measuring the intensity of the diffracted waves.
Especially, since intensities of the diffracted waves D+ in the +.theta. direction and those D- in the -.theta. direction varies in the opposite directions in response to the relative displacement in the direction (X direction) perpendicular to the grating within the surface thereof, so that in principle it becomes possible to measure the relative displacement in the X direction by measuring the difference in intensity between two diffracted waves I(D+)-I(D-).
However, the method described above with reference to FIG. 1 has defects so that it cannot be satisfactorily applied in practice because of the reasons described below. First, D+ and D- consist of many diffracted waves as shown in FIG. 1. The wave which is diffracted at the i-th order by the diffraction grating G1, diffracted at the j-th order by the diffraction grating G2 and further diffracted at the k-th order by the diffraction grating G1 is designated by D(i,j,k) while the wave which is reflected and diffracted at the i-th order on the upper surface of the diffraction grating G1 is designated by R(i). Then in the case of the first order diffraction, D+ is a mixed wave of diffracted waves R(1) and D(i,j,k) (where i+j+k=1) such as D(0,0,1), D(0,1,0), D(1,0,0), D(0,-1,2), D(-1,0,2) and so on. For the sake of simplicity, FIG. 1 only shows R(1), D(-1,1,1), D(0,0,1), D(1,-1,1) and D(1,0,0). The dependence of these diffracted waves on the relative displacement in the X direction of the gratings and on the distance S between the diffraction gratings G1 and G2 vary depending upon their diffraction orders. Thus, the intensities of D+ and D- are a complicated function of the relative displacement in the X direction and the distance S so that the measurement of the relative displacement in the X direction by the measurement of the difference between I(D+) and I(D-) is limited within an extremely limited range of the distance S.
Furthermore, if there exist differences in characteristics between the instruments for measuring the intensities of D+ and D-, the measured relative displacement includes errors. It follows therefore that in order to improve the accuracy of measurement, the characteristics of the instruments for measuring the intensities of D+ and D- must be made to strictly coincide with each other. As a result, the measurement with a high degree of accuracy becomes difficult.
FIG. 2 shows a measurement method capable of measuring the relative displacement without depending on the distance S, which is proposed by the present inventor and is disclosed in the Japanese Patent Application No. 60-165231. According to this method, two diffraction gratings G1 and G1' are formed on the object 1 and are spaced apart from each other by a suitable distance so that only the diffracted waves in the specific orders are incident on the diffraction grating G2 of the object 2, whereby the strong dependence of the intensity of the diffracted wave D from the diffraction grating G2 on the distance S between the diffraction gratings G1 and G2 is eliminated. Therefore, the relative displacement in the X direction can be measured by measuring the intensity of the diffracted light D by a detector 3 without being limited by the range of the distance S.
However, the above-described method also has the following defects so that it can not be satisfactorily used in practice. First, the intensity I(D) of the diffracted wave D depends on the displacement x in the x direction in proportion to cos.sup.2 (2.pi.x/d) (where d is the period of G1). But, the absolute value of I(D) is influenced by various factors so that it is impossible to theoretically estimate. It follows therefore in order to obtain x from the intensity I(D), the variation in I(D) must be measured while X is varied in the range of about d/4 in practice. Another defect of the above-described method is that since the intensity of the diffracted wave is used as a signal representative of a displacement (a displacement signal), the measurement result is easily adversely affected very often by the variations of characteristics of the measurement instrument. Furthermore, the intensity of the diffracted wave incident to the diffraction grating G2 from the diffraction gratings G1 and G1' varies in response to the variation in S so that the above-described measurement method has a further defect that the variation in S during the measurement process cannot be permitted, if it is desired to measure x with a high degree of accuracy.